Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension

TLDR

Stress singularities in plates under in‑plane extension are examined, extending prior bending‑case analyses. Three radial edge boundary conditions—free‑free, clamped‑clamped, and clamped‑free—are investigated. For vertex angles below 180°, singular stresses appear only with mixed clamped‑free conditions, with a stronger singularity than in bending; for angles between 180° and 360°, all cases can exhibit singularities; analogous characteristic equations arise when Poisson’s ratio is replaced by its negative, and the free‑free plate locally behaves like a clamped‑clamped bending plate regardless of Poisson’s ratio, implying the free‑free analysis applies to stress concentrations in V‑shaped notches.

Abstract

Abstract As an analog to the bending case published in an earlier paper, the stress singularities in plates subjected to extension in their plane are discussed. Three sets of boundary conditions on the radial edges are investigated: free-free, clamped-clamped, and clamped-free. Providing the vertex angle is less than 180 degrees, it is found that unbounded stresses occur at the vertex only in the case of the mixed boundary condition with the strength of the singularity being somewhat stronger than for the similar bending case. For vertex angles between 180 and 360 degrees, all the cases considered may have stress singularities. In amplification of some work of Southwell, it is shown that there are certain analogies between the characteristic equations governing the stresses in extension and bending, respectively, if ν, Poisson’s ratio, is replaced by −ν. Finally, the free-free extensional plate behaves locally at the origin exactly the same as a clamped-clamped plate in bending, independent of Poisson’s ratio. In conclusion, it is noted that the free-free case analysis may be applied to stress concentrations in V-shaped notches.